TY - JOUR
T1 - Asymptotic lifting for completely positive maps
AU - Forough, Marzieh
AU - Gardella, Eusebio
AU - Thomsen, Klaus
N1 - Publisher Copyright:
© 2024 The Author(s)
PY - 2024/12/15
Y1 - 2024/12/15
N2 - Let A and B be C⁎-algebras with A separable, let I be an ideal in B, and let ψ:A→B/I be a completely positive contractive linear map. We show that there is a continuous family Θt:A→B, for t∈[1,∞), of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then Θt can be chosen to have this property asymptotically. If A and B carry continuous actions of a second countable locally compact group G such that I is G-invariant and ψ is equivariant, we show that the family Θt can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that Θt is linear and completely positive for all t∈[1,∞). In the equivariant setting, if A, B and ψ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if G is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.
AB - Let A and B be C⁎-algebras with A separable, let I be an ideal in B, and let ψ:A→B/I be a completely positive contractive linear map. We show that there is a continuous family Θt:A→B, for t∈[1,∞), of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then Θt can be chosen to have this property asymptotically. If A and B carry continuous actions of a second countable locally compact group G such that I is G-invariant and ψ is equivariant, we show that the family Θt can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that Θt is linear and completely positive for all t∈[1,∞). In the equivariant setting, if A, B and ψ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if G is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.
KW - C-algebras
KW - Completely positive map
KW - Group action
KW - Lift
UR - http://www.scopus.com/inward/record.url?scp=85203415176&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2024.110655
DO - 10.1016/j.jfa.2024.110655
M3 - Journal article
AN - SCOPUS:85203415176
SN - 0022-1236
VL - 287
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 12
M1 - 110655
ER -